1. **Problem statement:** We have a production function $$y = x_1^{0.5} x_2^{0.25}$$ where $x_1$ and $x_2$ are inputs. 2. **Part a: Returns to scale** Returns to scale are determined by summing the exponents of inputs in the production function. 3. The sum of exponents is $$0.5 + 0.25 = 0.75$$. 4. Since $$0.75 < 1$$, the production function exhibits **decreasing returns to scale**. 5. **Part b: Profit-maximizing factor demand for $x_1$ in the short run** Given price of output $p$, prices of inputs $w_1$ and $w_2$, and fixed $x_2 = \bar{x}_2$. 6. The profit function is $$\pi = p y - w_1 x_1 - w_2 \bar{x}_2 = p x_1^{0.5} \bar{x}_2^{0.25} - w_1 x_1 - w_2 \bar{x}_2$$. 7. To maximize profit with respect to $x_1$, take the derivative and set to zero: $$\frac{d\pi}{dx_1} = p \cdot 0.5 x_1^{-0.5} \bar{x}_2^{0.25} - w_1 = 0$$. 8. Solve for $x_1$: $$p \cdot 0.5 x_1^{-0.5} \bar{x}_2^{0.25} = w_1$$ $$0.5 p \bar{x}_2^{0.25} = w_1 x_1^{0.5}$$ $$x_1^{0.5} = \frac{0.5 p \bar{x}_2^{0.25}}{w_1}$$ 9. Square both sides: $$x_1 = \left(\frac{0.5 p \bar{x}_2^{0.25}}{w_1}\right)^2$$ 10. **Part c: Short-run supply function** Supply is output $y$ as a function of price $p$ and fixed $\bar{x}_2$. 11. Substitute $x_1$ from step 9 into production function: $$y = x_1^{0.5} \bar{x}_2^{0.25} = \left(\left(\frac{0.5 p \bar{x}_2^{0.25}}{w_1}\right)^2\right)^{0.5} \bar{x}_2^{0.25} = \frac{0.5 p \bar{x}_2^{0.25}}{w_1} \bar{x}_2^{0.25}$$ 12. Simplify: $$y = \frac{0.5 p}{w_1} \bar{x}_2^{0.25 + 0.25} = \frac{0.5 p}{w_1} \bar{x}_2^{0.5}$$ **Final answers:** - a) Decreasing returns to scale - b) $$x_1^* = \left(\frac{0.5 p \bar{x}_2^{0.25}}{w_1}\right)^2$$ - c) $$y^* = \frac{0.5 p}{w_1} \bar{x}_2^{0.5}$$